Frequency Domain Analysis

Much in a similar way as to the typical Time Domain Analysis, where we analyze a system over a time scale (can also be said as time being the independent variable), in frequency domain analysis we analyze a system based on it’s response to $Asin(\omega t)$ signal inputs.

These can be expressed graphically both in: Nyquist Plot (Polar Locus) which displays our values of $G(j\omega )$ for a range of frequencies through the complex plane. Bode Diagram which displays the phase and magnitude in seperate plots both as functions of $\omega$

Example:

Given a system with a Transfer Function of:

$$G(s)=10\frac{(s+1)}{(s+2)(s+5)}$$

Purely by inspection we are able to determine we have singularities at $\{s=-1\}(zeros),\{s=-2,s=-5\}(poles)$ for a standard time domain analysis. Much in a similar way for our Frequency Domain Analysis we are able to deduce that we have singularities occurring at: $\{s=1rad/s\}(zeros),\{s=2rad/s,s=5rad/s\}(poles)$. This means that we are able to write our frequency response as such:

$$G(j\omega )=\frac{(j\omega +1)}{(\frac{j\omega }{2}+1)(\frac{j\omega }{5}+1)}$$

Performance Criteria

Some criteria for deciding upon a satisfactory closed-loop system can be formulated:

1. The closed loop is stable — the Nyquist Locus of the open-loop system must pass to the right of the $-1$ critical point.
2. The peak value of the closed-loop frequency response must not be “too high” which would indicate a resonance — the Nyquist Locus must not pass near the −1 critical point (e.g., Phase Margin $\approx 45\to 60^o$ ).
3. The Bandwidth of the closed-loop must be satisfactory as this determines the speed of response and the frequency range within which disturbances are rejected. The bandwidth of a system is defined as: the point at which the magnitude falls to $-3$dB ($\frac{1}{2}$). It can be shown that this corresponds to a magnitude of the open-loop transfer function of $0$dB ($1$).
4. The “steady-state error” to a “prototype” (step, ramp, etc) signal must be within some specified limit — recall that an increase of controller gain reduces this error.
5. The design should be “robust” against modelling errors or reasonable changes in system dynamics — the gain margin should be satisfactory.